The American goalkeeper has played under Moyes since he signed him from United in 2006 and he has been one of the key men at Everton since he arrived.
Howard admits Moyes is going to be a huge loss to Everton, but he will make a big impression at Old Trafford.
"By coming to work every day and not taking any short cuts. He does not allow players to rest or slacken off," said Howard.
"Look at a player like me, selected for damn near every game, so I think he has the utmost trust in me. But I can tell you with my hand on my heart that in training, when the goalkeepers get called over to work with the rest of the first team, I feel a bit of nerves.
"You might think it's only the training ground, but if I throw one in, it's not as if he looks the other way, he gives me a hammering from the touchline.
"After seven years you'd think a guy like me, who he trusts would get a break, but no, he is still on at you.
"That's his ethos, that's just how he is. He keeps you on your toes and believing you have to give more."
Howard hopes that Everton can be as successful as United in terms of finding a new manager.
"United and Everton are in the same boat with having to replace the manager in terms of expectation and all the rest. We will need to fill the vacancy with a very strong character. And I have no doubt that will happen," he said.
"His players have trusted in him and it's kind of a two-way street. It was a happy marriage for Moyes and for Everton.
"Everyone is saying United have to have faith in him, give him time to work his magic. Well it's the same with the new guy who comes here.
"We are a strong group of quality players and the foundations are set. The table is set for the next guy and I have no doubt if he is a strong character, a guy who can lead us well, we will have no problems."